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The problem describes three points \( A \), \( M \), and \( J \) on a line segment, where Madison (M) is located between Anoa (A) and Jamie (J). The distances are given as follows:

- Distance between Anoa's house and Madison's house: \( 3x + 2 \) miles.
- Distance between Madison's house and Jamie's house: \( 3x + 4 \) miles.
- Distance between Anoa's house and Jamie's house: \( 9x - 3 \) miles.

Since Madison is between Anoa and Jamie, we can combine the distances from \( A \) to \( M \) and from \( M \) to \( J \), which should equal the total distance from \( A \) to \( J \).

Thus, the equation to solve is:
\[
(3x + 2) + (3x + 4) = 9x - 3
\]

### Step-by-Step Solution:

1. Combine like terms on the left side of the equation:
   \[
   6x + 6 = 9x - 3
   \]

2. Subtract \( 6x \) from both sides:
   \[
   6 = 3x - 3
   \]

3. Add 3 to both sides:
   \[
   9 = 3x
   \]

4. Divide both sides by 3:
   \[
   x = 3
   \]

So, \( x = 3 \).

### Finding the Distance Between Madison and Jamie's House:

Now that we know \( x = 3 \), substitute it into the expression for the distance between Madison and Jamie:
\[
3x + 4 = 3(3) + 4 = 9 + 4 = 13 \, \text{miles}
\]

### Final Answer:
- \( x = 3 \)
- The distance between Madison and Jamie's house is 13 miles.

Would you like further details or have any questions?

Here are 5 related questions to expand your understanding:
1. How would the equation change if Madison was not between Anoa and Jamie?
2. Can you solve a similar problem where the total distance is expressed as \( 10x + 5 \)?
3. What happens to the total distance if \( x \) increases by 2?
4. How do you set up an equation when the points form a different shape, like a triangle?
5. What real-life situations could this kind of distance calculation be useful for?

**Tip:** When solving distance problems on a line, always make sure the sum of the parts equals the total distance to maintain consistency.

DISTANCE Madison lives between Anoa and Jamie as depicted on the line segment. The distance between Anoa’s house and Madison's house is represented by 3x + 2 miles, the distance between Madison's house and Jamie's house is represented by 3x + 4 miles, and the distance between Anoa's house and Jamie's house is represented by 9x - 3 miles. Find the value of x. Then find the distance between Madison's house and Jamie's house.

Learn how to solve a distance problem involving linear equations where Madison lives between Anoa and Jamie. Given the distances 3x + 2, 3x + 4, and 9x - 3, find the value of x and calculate the distance between Madison's and Jamie's houses. Suitable for Grades 7-9.

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